Why do people utter the phrase "bread and butter" when walking on the sidewalk?

It's a superstition.

""Bread and butter" is a superstitious blessing or charm, typically said by young couples or friends walking together when they are forced to separate by an obstacle, such as a pole or another person. By saying the phrase, the bad luck of letting something come between them is thought to be averted. Both walkers must say the phrase, and if they do not do this, then a bitter quarrel is expected to occur. The concept derives from the difficulty of separating butter from bread once it has been spread – buttered bread cannot be "unbuttered". Another phrase which may be used in this way is "salt and pepper.""

https://en.wikipedia.org/wiki/Bread_and_butter

Please note: this is probably an American superstition. I've never heard of it over here in the UK. (Side thought: why are there always superstitions and never ordinary ones?)

This is indeed an American superstition, and one this clueless American had never heard of. (I do know that if I step on a crack I'll break my mother's back.)

The footnotes on the Wikipedia article above go to:
"Signs and Superstitions Collected from American College Girls", The Journal of American Folklore
Louisiana folklore miscellany
101 American superstitions
Ozark tales and superstitions

As to ordinary stitions, just stand still and you've got one!

superstitious Look up superstitious at Dictionary.com
late 14c., from O.Fr. superstitieux, from L. superstitiosus, from superstitionem (nom. superstitio) "prophecy, soothsaying, excessive fear of the gods," perhaps originally "state of religious exaltation," related to superstes (gen. superstitis) "standing over or above," also "standing by, surviving," from superstare "stand on or over, survive," from super "above" (see super-) + stare "to stand," from PIE base *sta- "to stand" (see stet). There are many theories for the L. sense development, but none has yet triumphed. Superstition is attested from 1402. In Eng., originally especially of religion; sense of "unreasonable notion" is from 1794.
http://www.etymonline.com/index.php?term=superstitious

I had never heard of it either.

Then in two different episodes of my favorite show "The Twilight Zone" I saw it.

I grew up in Scotland having been taught "Bread and butter" by my mom. I've passed it on to my son so its also being used in South Africa!